FWIW, I have never needed to lube pistol caliber cases being sized in a TC die (I use RCBS), never had one jam or break.
Sometimes I will encounter a bunch of cases that require extraordinary effort to push into the sizer, but this being mostly range brass I make one of two assumptions about them: (1) case is excessively deformed due to previous owner loading them to high or excessively high pressures, or (2) the gun it was fired out of may have an out of spec chamber. Both of these conditions will cause the case to swell more than is the norm, and this will also lead to shortened case life or incipient case head separation (which is what happened in your die when the case broke).
I have no experience with Lee dies. It is also possible that the Lee dies are of a closer tolerance than the RCBS dies.
If this is your own brass and it is factory brass (new ammo) fired out of a new (or at least, not worn-out) gun, you can eliminate both possibilities noted above, and most likely it is then just a case of either plain old unreliable brass or more likely, the sizer die is of closer tolerance than whatever you are used to using before.
Now, since you already had a case head separation, a careful man would go back and inspect the sized cases for any signs of case head separation (including any stress lines in the case web). It's a pain but is OK if the case breaks in the sizer, it is not OK if it breaks in the chamber of your gun.
All that being said, there is no rule that says you should not use lube with TC dies. Personally, I use Hornady aerosol case lube, none of the pump-spray or liquid roll-it-on-a-pad lube. Used sparingly, that stuff will last quite a while.
Cheers!